2. Special Right Triangles
Directions
As you view this presentation, take
notes and work out the practice
problems.
When you get to the practice problem
screens, complete the step in your
notebook before continuing to the
next slide.
3. 45- 45- 90 Triangles
⢠A 45 â 45 â 90 triangle is also
known as an isosceles right
triangle.
⢠An isosceles right triangle is a
right triangle with 2 equal
sides or legs. (a = b)
⢠The 2 angles across from the
equal sides each measure 45o.
(angle A = angle B = 45o)
c
a
b
45o
45o
A
C B
4. 45- 45- 90 Triangles
⢠Because the lengths of the 2
legs in 45 â 45 â 90 triangle
are equal, the legs are usually
labeled x.
⢠The hypotenuse in a 45-45-90
triangle is often labeled h.
hx
x
45o
45o
5. 45- 45- 90 Triangles
Findingthe Length of the Hypotenuse
⢠The Pythagorean Theorem
can be used to find the length
of the hypotenuse when
given the length of the legs.
⢠x2 + x2 = h2
⢠2x2 = h2
⢠đĽ2 â 2 = h2
⢠x 2 = h
⢠You can save a lot of time and
work if you remember
h = x 2
hx
x
45o
45o
6. 45- 45- 90 Triangles
Practice Problem 1
⢠Find the length of x
h
x = ?
5
45o
45o
7. 45- 45- 90 Triangles
Practice Problem 1
⢠Find the length of x
⢠The two legs of a 45 â 45 â 90
triangle are equal so
x = 5
h
x = 5
5
45o
45o
8. 45- 45- 90 Triangles
Practice Problem 1
⢠Find the length of h
h = ?
x= 5
5
45o
45o
9. 45- 45- 90 Triangles
Practice Problem 1
⢠Find the length of h
⢠You can always use the
Pythagorean Theorem to find
the length of h.
⢠But if you remember the
shortcut
h = x 2
x = 5
5
45o
45o
10. 45- 45- 90 Triangles
Practice Problem 1
⢠Find the length of h
⢠You can always use the
Pythagorean Theorem to find
the length of h.
⢠But if you remember the
shortcut
⢠h = 5 2
h = 5 2
x = 5
5
45o
45o
11. 45- 45- 90 Triangles
Findingthe Lengths of the Legs
⢠The Pythagorean Theorem can be used
to find the lengths of the legs when
given the length of the hypotenuse.
⢠x2 + x2 = h2
⢠2x2 = h2
⢠x2 =
â2
2
⢠đĽ2 =
â2
2
⢠x =
â
2
*
2
2
=
â 2
2
⢠(Remember to always rationalize the
denominator)
hx
x
45o
45o
12. 45- 45- 90 Triangles
Findingthe Lengths of the Legs
You can save a lot of time
and work if you remember
x =
â 2
2 h
x =
â 2
2
x =
â 2
2
45o
45o
13. 45- 45- 90 Triangles
Practice Problem 2
⢠Find the length of x
h = 3
x = ?
x = ?
45o
45o
14. 45- 45- 90 Triangles
Practice Problem 2
⢠Find the length of x
⢠You can always use the
Pythagorean Theorem to find
the lengths of the legs.
h = 3
x = ?
x = ?
45o
45o
15. 45- 45- 90 Triangles
Practice Problem 2
⢠Find the length of x
⢠But if you remember the
shortcut
⢠x =
â 2
2
h = 3
x =
â 2
2
45o
45o
x =
â 2
2
16. 45- 45- 90 Triangles
Practice Problem 2
⢠Find the length of x
⢠But if you remember the
shortcut
⢠x =
â 2
2
⢠Then x =
3 2
2
h = 3
x =
3 2
2
45o
45o
x =
3 2
2
17. 45- 45- 90 Triangles
Practice Problem 3
⢠Find the length of x
h = 1
x = ?
x = ?
45o
45o
18. 45- 45- 90 Triangles
Practice Problem 3
⢠Find the length of x
⢠You can always use the
Pythagorean Theorem to find
the lengths of the legs.
h = 1
x = ?
x = ?
45o
45o
19. 45- 45- 90 Triangles
Practice Problem 3
⢠Find the length of x
⢠But if you remember the
shortcut
⢠x =
â 2
2
h = 1
x =
â 2
2
45o
45o
x =
â 2
2
20. 45- 45- 90 Triangles
Practice Problem 3
⢠Find the length of x
⢠But if you remember the
shortcut
⢠x =
â 2
2
⢠Then x =
1 2
2
=
2
2
h = 1
x =
2
2
45o
45o
x =
2
2
21. 45- 45- 90 Triangles
in the Unit Circle
⢠In the Unit Circle:
⢠h = 1
⢠So remembering this
shortcut for a 45 â 45 - 90
triangle will save you time
and work.
⢠x =
2
2
h = 1
x =
2
2
45o
45o
x =
2
2